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Descartes Total Angular Defect

The total angular defect is the sum of the Angular Defects over all Vertices of a Polyhedron, where the Angular Defect $\delta$ at a given Vertex is the difference between the sum of face angles and $2\pi$. For any convex Polyhedron, the Descartes total angular defect is

\Delta=\sum_i \delta_i = 4\pi.
\end{displaymath} (1)

This is equivalent to the Polyhedral Formula for a closed rectilinear surface, which satisfies
\end{displaymath} (2)

A Polyhedron with $N_0$ equivalent Vertices is called a Platonic Solid and can be assigned a Schläfli Symbol $\{p, q\}$. It then satisfies

\end{displaymath} (3)

\delta=2\pi-q\left({1-{2\over p}}\right)\pi,
\end{displaymath} (4)

N_0={4p\over 2p+2q-pq}.
\end{displaymath} (5)

See also Angular Defect, Platonic Solid, Polyhedral Formula, Polyhedron

© 1996-9 Eric W. Weisstein